Feynman Path Integral Formulation

22 1 Continuum FormulationFurthermore, in empty space T μν = 0, which then implies the vanishing of RiemannthereR μνρσ = 0 (1.114)As a result in three dimensions classical spacetime is locally flat everywhere outsidea source, gravitational fields do not propagate outside matter, and two bodies cannotexperience any gravitational force: they move uniformly on straight lines.There cannot be any gravitational waves either: the Weyl tensor which carries informationabout gravitational fields not determined locally by matter, vanishes identicallyin three dimensions. One further surprising (and disappointing) conclusionfrom the previous arguments is that black holes cannot exist in three dimensions,as spacetime is flat outside matter, which always allows light emitted from a starto escape to infinity. One interesting case though is three-dimensional anti-DeSitterspace, with a scaled cosmological constant λ = −1/ξ 2 . There one can show thatobjects which could be described as black holes exist, with a black hole horizonin the non-rotating case at r 0 = ξ √ MG, where M is the mass of the collapsed object(Banados Teitelboim and Zanelli, 1992). Note that in three dimensions G hasdimensions of a length, so that the product MG ends up being dimensionless. Thescale of the horizon is therefore supplied by the scale ξ .What seems rather puzzling at first is that Newtonian theory seems to make perfectsense in d = 3. The Newtonian potential is non-vanishing and grows logarithmicallywith distance,∫V (r) ∝ G d 2 ke ix·k /k 2 ∼ G logr . (1.115)This can only mean that the Newtonian theory is not recovered in the weak field limitof the relativistic theory (Deser, Jackiw and Templeton, 1982; Giddings, Abbott andKuchar, 1984).To see this explicitly, it is sufficient to consider the trace-reversed form of thefield equations,(R μν = 8πG T μν − 1 )d − 2 g μν T , (1.116)with T = T λ λ , in the weak field limit. In the linearized theory, with h μν = g μν −η μν ,and in the Hilbert-DeDonder gaugeone obtains the wave equation✷h μν∇ λ h λ μ − 1 2 ∇ μh λ λ = 0 , (1.117)= −16πG(τ μν − 1 )d − 2 η μν τ, (1.118)with τ μν the linearized stress tensor. After neglecting the spatial components of τ μνin comparison to the mass density τ 00 , and assuming that the fields are quasi-static,one obtains a Poisson equation for h 00 ,